Other Special Functions

ArithmeticGeometricMean(x, y) : FldPrElt, FldPrElt -> FldPrElt

AGM(f, g) : RngSerElt, RngSerElt -> RngSerElt

Return the hyperbolic arithmetic-geometric mean of the series f and g defined over a field. The valuations of f and g must be equal.

ArithmeticGeometricMean(x, y) : FldPrElt, FldPrElt -> FldPrElt

AGM(x, y) : FldPrElt, FldPrElt -> FldPrElt

Returns the arithmetic-geometric mean of the free real or complex numbers x and y, defined as the limit of either of the sequences x_i, y_i where x_0 = x, y_0 = y and x_(i + 1) = (x_i + y_i)/2, y_(i + 1) = Sqrt(x_i y_i). The function calculates both sequences, and when the numbers are within the desired precision of each other, it returns one of them.

BernoulliNumber(n) : RngIntElt -> FldRatElt

For a non-negative integer n, return the value of the n-th Bernoulli number B_n, defined by (t/e^t - 1)=sum_(n=0)^Infinity B_n(t^n/n!).

BernoulliApproximation(n) : RngIntElt -> FldPrElt

For a non-negative integer n, return an approximation in the field of free real numbers to the value of the n-th Bernoulli number B_n, defined by (t/e^t - 1)=sum_(n=0)^Infinity B_n(t^n/n!).

DawsonIntegral(r) : FldReElt -> FldReElt

Given a real number r, compute the value of Dawson's integral, e^(-x^2) times the integral from 0 to x of e^(u^2) with respect to u, at x = r.

ErrorFunction(r) : FldReElt -> FldReElt

Erf(r) : FldReElt -> FldReElt

Given a real number r, calculate the value of the error function erf. This is the value of the square root of (4/pi) times the integral from 0 to x of e^(-u^2) with respect to u, at x = r for r>0, and for r<0 it is defined by erf(x)= - erf( - x), while erf(0)=0.

Here r is allowed to be free or of fixed precision, and the result will be in the same field.

ComplementaryErrorFunction(r) : FldReElt -> FldReElt

Erfc(r) : FldReElt -> FldReElt

Given a real number r, calculate the value of the complementary error function. This is the value of y = ( erfc)(x) = 1 - erf(x). for the error function erf as defined above.

Here r is allowed to be free or of fixed precision, and the result will be in the same field.

ExponentialIntegral(r) : FldReElt -> FldReElt

Given a real number r, calculate the value of the exponential integral, that is, the principal value of the integral from minus infinity to x of e^u / u with respect to u at x = r.

Here r is allowed to be free or of fixed precision, and the result will be in the same field.

ExponentialIntegralE1(r) : FldReElt -> FldReElt

Given a free real number r, calculate the value of the exponential integral E1, that is, the principal value of the integral from x to infinity of e^u / u with respect to u at x = r.

LogIntegral(s) : FldPrElt -> FldPrElt

LogIntegral(r) : FldReElt -> FldReElt

Given a non-negative real number r that is not equal to 1, evaluate the logarithmic integral y = li(x) at x = r. This integral is defined to be the principal value of the integral from 0 to x of 1 / log(u) with respect to u.

ZetaFunction(s) : FldPrElt -> FldPrElt

ZetaFunction(R, n) : FldRe, RngIntElt -> FldReElt

These functions calculate values of the Riemann zeta-function, which is the analytic continuation of zeta(z) = the sum from i equals 1 to infinity of 1 / i^n (convergent for Re(z)> 1). The version with one argument takes a free real or complex number r != 1 and returns a free real or complex number.

The version with two arguments is much more restricted; it takes a real field R of fixed precision and an integer n != 1, and returns zeta(n) in R.

For non-integer arguments the Euler-Maclaurin series is used, for non-positive and even positive integers the Bernoulli numbers are used. The fixed precision version uses Euler-Maclaurin for odd positive integers as well, the free version uses modular forms.